On large bipartite graphs of diameter 3

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers d ≥ 2 and D ≥ 2 , find the maximum number N b ( d , D ) of vertices in a bipartite graph of maximum degree d and diameter D . In this context, the bipartite Moore bound M b ( d , D ) represents a general upper bound for N b ( d , D ) . Bipartite graphs of order M b ( d , D ) are very rare, and determining N b ( d , D ) still remains an open problem for most ( d , D ) pairs. aper is a follow-up of our earlier paper (Feria-Purón and Pineda-Villavicencio, 2012 [5]), where a study on bipartite ( d , D , − 4 ) -graphs (that is, bipartite graphs of order M b ( d , D ) − 4 ) was carried out. Here we first present some structural properties of bipartite ( d , 3 , − 4 ) -graphs, and later prove that there are no bipartite ( 7 , 3 , − 4 ) -graphs. This result implies that the known bipartite ( 7 , 3 , − 6 ) -graph is optimal, and therefore N b ( 7 , 3 ) = 80 . We dub this graph the Hafner–Loz graph after its first discoverers Paul Hafner and Eyal Loz. proach here presented also provides a proof of the uniqueness of the known bipartite ( 5 , 3 , − 4 ) -graph, and the non-existence of bipartite ( 6 , 3 , − 4 ) -graphs. ition, we discover at least one new largest known bipartite–and also vertex-transitive–graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for N b ( 11 , 3 ) .